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This paper presents a reliable approach for solving linear systems of ordinary and fractional differential equations. First, the FDEs or ODEs of a system with initial conditions to be solved are transformed to Volterra integral equations. Then Taylor expansion for the unknown function and integration method are employed to reduce the resulting integral equations to a new system of linear equations for the unknown and its derivatives. The fractional derivatives are considered in the Riemann-Liouville sense. Some numerical illustrations are given to demonstrate the effectiveness of the proposed method in this paper. [ABSTRACT FROM AUTHOR]

The power graph $${\fancyscript{P}}(G)$$ of a group $$G$$ is the graph which has a vertex set of the group elements and two elements are adjacent if one is a power of the other. Chakrabarty, Ghosh, and Sen proved the main properties of the undirected power graph of a finite group. The aim of this paper is to generalize some results of their work and presenting some counterexamples for one of the problems raised by these authors. It is also proved that the power graph of a $$p$$ -group is $$2$$ -connected if and only if the group is a cyclic or generalized quaternion group and if $$G$$ is a nilpotent group which is not of prime power order then the power graph $${\fancyscript{P}}(G)$$ is $$2$$ -connected. We also prove that the number of edges of the power graph of the simple groups is less than or equal to the number of edges in the power graph of the cyclic group of the same order. This partially answers to a question in an earlier paper. Finally, we give a complete classification of groups in which the power graph is a union of complete graphs sharing a common vertex. [ABSTRACT FROM AUTHOR]

Björner, Lovász and Shor introduced a chip-firing game on a finite graph $$G$$ as follows. We put some chips on each vertex of $$G$$ , we say that a vertex is ready if it has at least as many chips as its degree, in which case we can fire it and the result is that it distributes one chip to each of its neighbors, this may cause other vertices to be ready, and so on. This game continues until no vertex can be fired. In this paper, we study chip-firing games on complete graphs. We obtain a sufficient and necessary condition for chip-firing games on complete graphs to be finite. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a concept of graph convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping in Banach spaces and prove an equivalence theorem between graph convergence and resolvent operator convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping. Further, we consider a system of generalized variational inclusions involving $$H(\cdot ,\cdot )$$ -co-accretive mapping in real $$q$$ -uniformly smooth Banach spaces. Using resolvent operator technique, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions under some suitable conditions. Further, we discuss the convergence of iterative algorithm using the concept of graph convergence. Our results can be viewed as a refinement and generalization of some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we use a result of N. S. Feldman to show that there are no supercyclic subnormal tuples in infinite dimensions. Also, we investigate some spectral properties of hypercyclic tuples of operators. Besides, we prove that if $$T$$ is a supercyclic $$\ell $$ -tuple of commuting $$n\times n$$ complex matrices, then $$\ell \ge n$$ and also there exists a supercyclic $$n$$ -tuple of commuting diagonal $$n\times n$$ matrices. Furthermore, we see that if $$T=(T_{1},\ldots ,T_{n})$$ is a supercyclic $$n$$ -tuple of commuting $$n\times n$$ complex matrices, then $$T_{j}$$ 's are simultaneously diagonalizable. [ABSTRACT FROM AUTHOR]

In the present paper, we study the extrinsic and intrinsic geometry of submanifolds of an almost contact metric manifold admitting a quarter-symmetric metric connection. We deduce Gauss, Codazzi and Ricci equations corresponding to the quarter-symmetric metric connection and show some applications of these equations. Finally, we give an example verifying the results. [ABSTRACT FROM AUTHOR]

Let $$G$$ be a graph with degree sequence $$d_1,d_2,\ldots ,d_n$$ . Given a positive integer $$p$$ , denote by $$e_p(G)=\sum _{i=1}^n d_i^p$$ . Caro and Yuster introduced a Turán-type problem for $$e_p(G)$$ : given an integer $$p$$ , how large can $$e_p(G)$$ be if $$G$$ has no subgraph of a particular type. They got some results for the subgraph of particular type to be a clique of order $$r+1$$ and a cycle of even length, respectively. Denote by $$ex_p(n,H)$$ the maximum value of $$e_p(G)$$ taken over all graphs with $$n$$ vertices that do not contain $$H$$ as a subgraph. Clearly, $$ex_1(n,H)=2ex(n,H)$$ , where $$ex(n,H)$$ denotes the classical Turán number. In this paper, we consider $$ex_p(n, C_5)$$ and prove that for any positive integer $$p$$ and sufficiently large $$n$$ , there exists a constant $$c=c(p)$$ such that the following holds: if $$ex_p(n, C_5)=e_p(G)$$ for some $$C_5$$ -free graph $$G$$ of order $$n$$ , then $$G$$ is a complete bipartite graph having one vertex class of size $$cn+o(n)$$ and the other $$(1-c)n+o(n)$$ . [ABSTRACT FROM AUTHOR]

Let $${\mathcal {F}}$$ denote the class of all functions univalent in the unit disk $$\Delta \equiv \{\zeta \in {\mathbb {C}}\,:\,\left| \zeta \right| <1\}$$ and convex in the direction of the real axis. The paper deals with the subclass $${\mathcal {F}}^{(n)}$$ of these functions $$f$$ which satisfy the property $$f(\varepsilon z)=\varepsilon f(z)$$ for all $$z\in \Delta $$ , where $$\varepsilon =e^{2\pi i/n}$$ . The functions of this subclass are called $$n$$ -fold symmetric. For $${\mathcal {F}}^{(n)}$$ , where $$n$$ is odd positive integer, the following sets, $$\bigcap _{f\in {\mathcal {F}}^{(n)}} f(\Delta )$$ -the Koebe set and $$\bigcup _{f\in {\mathcal {F}}^{(n)}} f(\Delta )$$ -the covering set, are discussed. As corollaries, we derive the Koebe and the covering constants for $${\mathcal {F}}^{(n)}$$ . [ABSTRACT FROM AUTHOR]

There are many articles in the literature dealing with the first-order and the second-order differential subordination and differential superordination problems for analytic functions in the unit disk, but there are only a few articles dealing with the third-order differential subordination problems. The concept of third-order differential subordination in the unit disk was introduced by Antonino and Miller, and studied recently by Tang and Deniz. Let $$\Omega $$ be a set in the complex plane $$\mathbb {C}$$ , let $$\mathfrak {p}(z)$$ be analytic in the unit disk $$\mathbb {U}=\{z:z\in \mathbb {C}\quad \text {and} \quad |z|<1\}$$ , and let $$\psi : \mathbb {C}^4\times \mathbb {U}\rightarrow \mathbb {C}$$ . In this paper, we investigate the problem of determining properties of functions $$\mathfrak {p}(z)$$ that satisfy the following third-order differential superordination: As applications, we derive some third-order differential superordination results for analytic functions in $$\mathbb {U}$$ , which are associated with a family of generalized Bessel functions. The results are obtained by considering suitable classes of admissible functions. New third-order differential sandwich-type results are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the congruence lattice of a symmetric extended De Morgan algebra $$L$$ . We show that the congruence lattice of the algebra $$L$$ is a pseudocomplemented lattice, and that such a congruence lattice is a Stone lattice if and only if the lattice of the compact congruences on $$L$$ forms a complete Boolean lattice. In particular, we prove that the congruence lattice of $$L$$ is a Boolean lattice if and only if, it is a relative Stone lattice, which is the case, if and only if $$L$$ is finite. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the central limit theorem and the law of the iterated logarithm for Galton-Watson branching processes with time-dependent immigration in varying environments. [ABSTRACT FROM AUTHOR]